Solve 14 (4x - 3) ^ 2 + (4x + 3) ^ 2 = 25 | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$( 4 x - 3 ) ^ { 2 } + ( 4 x + 3 ) ^ { 2 } = 25$$

Solve for x

$x=\frac{\sqrt{14}}{8}\approx 0.467707173$
$x=-\frac{\sqrt{14}}{8}\approx -0.467707173$

Steps by Finding Square Root

Steps Using the Quadratic Formula

Steps by Finding Square Root

Use binomial theorem $\left(a-b\right)^{2}=a^{2}-2ab+b^{2}$ to expand $\left(4x-3\right)^{2}$.

$$16x^{2}-24x+9+\left(4x+3\right)^{2}=25$$

Use binomial theorem $\left(a+b\right)^{2}=a^{2}+2ab+b^{2}$ to expand $\left(4x+3\right)^{2}$.

$$16x^{2}-24x+9+16x^{2}+24x+9=25$$

Combine $16x^{2}$ and $16x^{2}$ to get $32x^{2}$.

$$32x^{2}-24x+9+24x+9=25$$

Combine $-24x$ and $24x$ to get $0$.

$$32x^{2}+9+9=25$$

Add $9$ and $9$ to get $18$.

$$32x^{2}+18=25$$

Subtract $18$ from both sides.

$$32x^{2}=25-18$$

Subtract $18$ from $25$ to get $7$.

$$32x^{2}=7$$

Divide both sides by $32$.

$$x^{2}=\frac{7}{32}$$

Take the square root of both sides of the equation.

$$x=\frac{\sqrt{14}}{8}$$ $$x=-\frac{\sqrt{14}}{8}$$

Steps Using the Quadratic Formula

Use binomial theorem $\left(a-b\right)^{2}=a^{2}-2ab+b^{2}$ to expand $\left(4x-3\right)^{2}$.

$$16x^{2}-24x+9+\left(4x+3\right)^{2}=25$$

Use binomial theorem $\left(a+b\right)^{2}=a^{2}+2ab+b^{2}$ to expand $\left(4x+3\right)^{2}$.

$$16x^{2}-24x+9+16x^{2}+24x+9=25$$

Combine $16x^{2}$ and $16x^{2}$ to get $32x^{2}$.

$$32x^{2}-24x+9+24x+9=25$$

Combine $-24x$ and $24x$ to get $0$.

$$32x^{2}+9+9=25$$

Add $9$ and $9$ to get $18$.

$$32x^{2}+18=25$$

Subtract $25$ from both sides.

$$32x^{2}+18-25=0$$

Subtract $25$ from $18$ to get $-7$.

$$32x^{2}-7=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $32$ for $a$, $0$ for $b$, and $-7$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$x=\frac{0±\sqrt{0^{2}-4\times 32\left(-7\right)}}{2\times 32}$$

Square $0$.

$$x=\frac{0±\sqrt{-4\times 32\left(-7\right)}}{2\times 32}$$

Multiply $-4$ times $32$.

$$x=\frac{0±\sqrt{-128\left(-7\right)}}{2\times 32}$$

Multiply $-128$ times $-7$.

$$x=\frac{0±\sqrt{896}}{2\times 32}$$

Take the square root of $896$.

$$x=\frac{0±8\sqrt{14}}{2\times 32}$$

Multiply $2$ times $32$.

$$x=\frac{0±8\sqrt{14}}{64}$$

Now solve the equation $x=\frac{0±8\sqrt{14}}{64}$ when $±$ is plus.

$$x=\frac{\sqrt{14}}{8}$$

Now solve the equation $x=\frac{0±8\sqrt{14}}{64}$ when $±$ is minus.

$$x=-\frac{\sqrt{14}}{8}$$

The equation is now solved.

$$x=\frac{\sqrt{14}}{8}$$ $$x=-\frac{\sqrt{14}}{8}$$